sum from n=1 to infinity of 5/(n^2+16)

{ "query": { "display": "$$\\sum_{n=1}^{\\infty\\:}\\frac{5}{n^{2}+16}$$", "symbolab_question": "BIG_OPERATOR#\\sum _{n=1}^{\\infty }\\frac{5}{n^{2}+16}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Series", "subTopic": "Convergence", "default": "\\mathrm{converges}" }, "steps": { "type": "interim", "title": "Check convergence of $$\\sum_{n=1}^{\\infty\\:}\\frac{5}{n^{2}+16}:{\\quad}$$converges", "input": "\\sum_{n=1}^{\\infty\\:}\\frac{5}{n^{2}+16}", "steps": [ { "type": "step", "primary": "Apply the constant multiplication rule: $$\\sum{c\\cdot{a_{n}}}=c\\cdot\\sum{a_{n}}$$", "result": "=5\\cdot\\:\\sum_{n=1}^{\\infty\\:}\\frac{1}{n^{2}+16}" }, { "type": "interim", "title": "Apply Series Comparison Test:$${\\quad}$$converges", "input": "\\sum_{n=1}^{\\infty\\:}\\frac{1}{n^{2}+16}", "steps": [ { "type": "definition", "title": "Comparison Test:", "text": "Let $$\\sum\\:a_n,\\:\\sum\\:b_n\\:$$be two positive sequences such that for all $$n,\\:\\quad\\:a_n\\le\\:b_n$$<br/>If $$\\sum\\:b_n\\:$$converges, so does $$\\sum\\:a_n$$<br/>If $$\\sum\\:a_n\\:$$diverges, so does $$\\sum\\:b_n$$", "secondary": [ "$$\\sum_{n=1}^{\\infty\\:}\\frac{1}{n^{2}+16}\\le\\:\\sum_{n=1}^{\\infty\\:}\\frac{1}{n^{2}}$$" ] }, { "type": "interim", "title": "Check convergence of $$\\sum_{n=1}^{\\infty\\:}\\frac{1}{n^{2}}:{\\quad}$$converges", "input": "\\sum_{n=1}^{\\infty\\:}\\frac{1}{n^{2}}", "steps": [ { "type": "interim", "title": "Apply p-Series Test:$${\\quad}$$converges", "input": "\\sum_{n=1}^{\\infty\\:}\\frac{1}{n^{2}}", "steps": [ { "type": "definition", "title": "p-Series Test:", "text": "If the series is of the form $$\\sum_{n=1}^{\\infty}\\frac{1}{n^p},\\:$$where $$p\\gt0$$<br/>$${\\quad}$$If $$p>1,\\:$$then the p-series converges<br/>$${\\quad}$$If $$0<p\\le1,\\:$$then the p-series diverges" }, { "type": "step", "primary": "$$p=2,\\:\\:p>1,\\:$$by the p-Series test criteria", "result": "=\\mathrm{converges}" } ], "meta": { "interimType": "Series Apply P Series Test 0Eq" } }, { "type": "step", "result": "=\\mathrm{converges}" } ], "meta": { "solvingClass": "Series", "interimType": "Sum Convergence Check 1Eq" } }, { "type": "step", "primary": "By the comparison test", "result": "=\\mathrm{converges}" } ], "meta": { "interimType": "Series Apply Comparison Test 0Eq" } }, { "type": "step", "result": "=5\\mathrm{converges}" }, { "type": "step", "result": "=\\mathrm{converges}" } ], "meta": { "solvingClass": "Series", "practiceLink": "/practice/series-practice#area=main&subtopic=Comparison%20Test", "practiceTopic": "Series Comparison Test" } } }